nLab class S field theory

Overview

Class $\mathcal{S}$ (for “six”) originally denoted a specific class of $N=2$ 4d super Yang-Mills theory introduced in Gaiotto, Moore & Neitzke 2009. They originate from KK-compactification of M5-branes on punctured Riemann surfaces $C$ and are labelled by a simply laced Lie group?, a Riemann surface $C$, and a decoration of the punctures of $C$ by defect operators.

By further compactifications, one relates theories of this class with certain superconformal field theories in dimension $2$ (or, in other formalism, chiral algebras) which are now also said to be of class $S$ (see at AGT correspondence).

References

Original articles

• Davide Gaiotto, Greg Moore, Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics 234 (2013) 239–403 arXiv:0907.3987 doi

see also

• Zafrir 2016.

In the AGT correspondence:

• Bruno Le Floch, A slow review of the AGT correspondence, J. Phys. A: Math. Theor. 55 (2022) 353002 doi arXiv:2006.14025

On vertex operator algebras/chiral algebras of class $\mathcal{S}$:

• Tomoyuki Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties, arXiv:1811.01577

  We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in physics. We show that there is a unique family of vertex algebras satisfying the required conditions and show that they are all simple and conformal. In fact, our construction works for any complex semisimple group G that is not necessarily simply laced. Furthermore, we show that the associated varieties of these vertex algebras are exactly the genus zero Moore-Tachikawa symplectic varieties that have been recently constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian for the Langlands dual group.

On superconformal theories of class $\mathcal{S}$:

• Leonardo Rastelli, S. S. Razamat, The superconformal index of theories of class $\mathcal{S}$, in: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer 2015 doi

• Christopher Beem, Wolfger Peelaers, Leonardo Rastelli, Balt C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 doi arXiv:1408.6522

On quantum Seiberg-Witten curves in relation to class S-theories and “M3”-defect branes inside M5-branes:

• Jin Chen, Babak Haghighat, Hee-Cheol Kim, Marcus Sperling, Elliptic Quantum Curves of Class $\mathcal{S}_k$, J. High Energ. Phys. 2021 28 (2021) [arXiv:2008.05155, doi:10.1007/JHEP03(2021)028]

category: physics